Before continuing presenting our results, let us say what exactly are we looking at when studying a long
sequence of auctions played by software agents. While one might hope that there is some kind of
convergence to fixed bids or distributions, it is well known that
regret-minimization dynamics in many games need not converge [1]}, [2]}, [3]}, [4]}.
For example, in our context, Figure REF  shows the dynamics of a simulation
of a generalized first-price auction for two slots where the two bidders have the same value.
(Throughout the dynamics, both agents bid nearly the same value and thus only a single bid-line is visible.)
As one may clearly see, the dynamics do not converge. What we wish to study
is what is important to the auctioneer and to the
bidders: the long-term distribution of the tuple of
bids. This distribution determines the average winning frequencies and
prices which in turn determine the users' utilities and the auctioneer's revenue
over the whole sequence of
auctions. For example,
Figure REF  shows a density map of
the empirical pair of bids in this simulation.
